The mathematics of disaster

Extreme events, such as tsunamis or network crashes, are by their very nature rare. But because they occur in a wide range of dynamic systems – from computers to climate – they are also, from one perspective, ubiquitous.

Probing for similarities between extreme events in a variety of arenas has led two researchers from the Department of Mechanical Engineering at the Massachusetts Institute of Technology, US, to uncover a deep mathematical architecture that might open the way to more efficient disaster warning systems.

In a report published in the journal Science Advances, Mohammad Farazmand and Themistoklis Sapsis outline a mechanism for identifying key patterns that occur in multi-dimensional, dynamic environments before an extreme rogue event.

Existing attempts to construct event prediction models, logically enough, proceed from the assumption that the physics of a system, expressed as dynamic equations, can be used to identify its initial conditions. Moving forward from those start-points can then yield predictions about future events.

However, as Sapsis and Farazmand, write, the complex physics of turbulent systems is often not fully observable, and calculations thus founder, with equations producing prediction outcomes that are not consistent with real world phenomena. 

“If we just blindly take the equations and start looking for initial states that evolve to extreme events, there is a high probability we will end up with initial states that are very exotic, meaning they will never ever occur for any practical situation,” Sapsis says. “So equations contain more information than we really need.”

To reduce the number of unrealistic variables in their model, the researchers combed through data gathered from actual extreme events, such as earthquakes and system failures, and identified actual precursors.

However, because such events are numerically rare, reaching reliable findings would have required a vast and practically unfeasible dataset. 

To offset this problem, Sapsis and Farazmand constructed a general mathematical framework, added in real world data, and devised an algorithm to chew through the results.

“We are looking at the equations for possible states that have very high growth rates and become extreme events, but they are also consistent with data, telling us whether this state has any likelihood of occurring, or if it’s something so exotic that, yes, it will lead to an extreme event, but the probability of it occurring is basically zero,” Sapsis says.

They then tested the model on a fluid dynamic flow that would describe a chaotic system, such as a plume of cigarette smoke or airflow around a moving wingtip, and successfully identified several precursor signals that described the beginning of an extreme event.

Further testing found that identifying the precursor signals was an accurate predictor of an extreme event between 75% and 99% of the time – the variation reflecting relative complexities of the systems tested.

The researchers now plan to apply their model to some real world situations in which extreme events can occur. Specifically, they intend to look at airflows around jet liners and ocean currents hitting oil platforms.

“This happens in random places around the world, and the question is being able to predict where these vortices or hotspots of extreme events will occur,” Sapsis says.

“If you can predict where these things occur, maybe you can develop some control techniques to suppress them.”